Burau Representation and Random Walk on String Links

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. §1. Beginning of the story: Burau representation The Burau matrices βi, 1 ≤ i ≤ n− 1, are n× n matrices given as follows: Let t 6= 0, 1 be a complex number. If we think of βi as a linear transformation on C , then βi = (1)⊕ · · · ⊕ (1) } {{ } i−1 copies ⊕ ( 1− t t 1 0 ) ⊕ (1)⊕ · · · ⊕ (1) } {{ } n−i−1 copies , and β i = (1)⊕ · · · ⊕ (1) } {{ } i−1 copies ⊕ ( 0 1 t̄ 1− t̄ ) ⊕ (1)⊕ · · · ⊕ (1) } {{ } n−i−1 copies . Here we use t̄ to denote t for simplicity. It is easy to check that βiβi+1βi = βi+1βiβi+1, βiβj = βjβi for |i− j| ≥ 2. Thus, sending the standard generators σi of the braid group Bn to βi defines the (non-reduced) Burau representation of Bn. For a given braid, its image under the Burau representation will be called the Burau matrix of that braid. There is an extensive literature on the Burau representation. We only mention an article of John Moody [7] where it was proved (settling a question of long time) that the Burau representation is not faithful. In [5], Vaughan Jones offered a probabilistic interpretation of the Burau representation. We quote from [3] (with a small correction): Wang is supported by an NSF Postdoctoral Fellowship. 2 XIAO-SONG LIN, FENG TIAN AND ZHENGHAN WANG “For positive braids there is also a mechanical interpretation of the Burau matrix: Lay the braid out flat and make it into a bowling alley with n lanes, the lanes going over each other according to the braid. If a ball traveling along a lane has probability 1− t of falling off the top lane (and continuing in the lane below) at every crossing then the (i, j) entry of the (non-reduced) Burau matrix is the probability that a ball bowled in the ith lane will end up in the jth.” Let us now consider string links. This notion was first introduced in [4]. We will generalize it a little bit here and still call the generalization string links. Essentially, a string link is an oriented tangle diagram (or simply a tangle) in the strip R × [0, 1] with bottom ends {1× 0, 2× 0, . . . , n× 0} (call them sources) and top ends {1 × 1, 2 × 1, . . . , n × 1} (call them sinks). There are exactly n strands, each of them giving an oriented path from a bottom source i×0 to a top sink j×1. See Figure 1. Two such string links are thought to be the same if they differ by a finite sequence of Reidermeister moves. Naturally, the set Sn of all string links with n strands has a semigroup structure such that Bn ⊂ Sn is a subgroup. T Figure 1. A favorable string link. We now define a representation of the semigroup Sn generalizing the Burau representation. We will assign to each element in Sn an n×n matrix whose entries are rational functions in t. Such an assignment will be multiplicative on Sn so that we get a representation of the semigroup Sn into the semigroup of n× n matrices. Starting at the point i × 0, we will try to walk up along strands of the given string link σ to get to the point j × 1 according to the the following rules: (1) The walking direction should always be in agreement with the orientation of strands of σ. (2) If we come to a crossing on the lower segment, we will keep walking on the lower segment passing through that crossing. (3) If we come to a crossing on the upper segment, we may choose either to jump down walking on the lower segment or keep walking on the upper segment passing through that crossing. Such a way of walking from i× 0 to j × 1 on σ is called a path. A loop is a part of a path along which we may come back to where we start on a string link. A path is called simple if it contains no loops. Obviously, there are only finitely many simple paths on a string link. A loop is simple if it contains no other loops except itself. There are only finitely many simple loops on a string link. Any path can be reduced down to a simple path by dropping off simple loops it contains. Therefore, there are at most countably many paths on a string link. Let us assign a weight w(P ) to a path P . Along P , there are many places where decisions are made about whether we jump down or keep walking. We will have a BURAU REPRESENTATION AND RANDOM WALK ON STRING LINKS 3 state at each of these places along P and w(P ) is the product of all states on that path. The states are determined as follows: (1) if we come to a positive crossing on the upper segment, the state is 1− t if we choose to jump down and t otherwise; and (2) if we come to a negative crossing on the upper segment, the state is 1− t̄ if we choose to jump down and t̄ otherwise. With all these said, the (i, j) entry of the n × n matrix assigned to the string link σ ∈ Sn is (1.1) ∑